Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations

Pages: 3733 - 3771, Volume 32, Issue 10, October 2012

doi:10.3934/dcds.2012.32.3733       Abstract        References        Full Text (632.3K)       Related Articles

Vedran Sohinger - Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States (email)

Abstract: In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we had previously used in [49, 50] to study the analogous problem in one dimension. Since we are working in two dimensions, a more detailed analysis of the resonant frequencies is needed, as was previously used in the work of Colliander-Keel-Staffilani-Takaoka-Tao [19].

Keywords:  Hartree equation, nonlinear Schrödinger equation, growth of high Sobolev norms, resonant decomposition.
Mathematics Subject Classification:  35Q55.

Received: May 2010;      Revised: January 2012;      Published: May 2012.

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